Q-centroid

As we have discussed in Section 2.2 and shall treat more precisely in Chapter 5, the coupling matrix element (3.2) for nonradiative transition between the Bom-Oppen-heimer adiabatic states can be calculated at a configuration qo, the so-called q-centroid. This permits Equation 3.2 to be rewritten in the form... 

As one should expect from the terminology, the earliest application of the q-centroid, strictly speaking, the r-centroid method was made in diatomic spectroscopy [131], although in a completely different physical sense. The q-centroid approximation presents a nontrivial generalization of the r-centroid approach of diatomic spectroscopy to the case of the nonradiative decay of polyatomic molecules. The importance of studying the q-dependence of the nonadiabatic coupling element was emphasized by Freed and Gelbart [132] and Freed and lin [27]. 

The q-centroid is introduced at the next level of approximation. Taking first the third-order terms occurring on the right (ket) side of (5.3), we have... 

For q being a promoting mode, only the part of (5.42) that contains qo i K t) contributes to the q-centroid. 

For optically allowed transitions, the electronic matrix element ( ) in (6.10) is usually a slowly varying function of the nuclear coordinates over the range where Xs Xo significantly differs from zero. Therefore, it may be removed from the inner element of the nuclear integration by using the q-centroid-type approximation. To obtain (6.10), we have assumed that the electric dipole transition is allowed for the decay from state t )j to the ground state and set = 1 in Equation 6.7. This is an excellent approximation for atomic transitions, since called long-wave approximation. 

Third, as a consequence of the foregoing, the use of IDs is suitable in investigating the dependence of the electronic matrix element for radiationless transition on the nuclear coordinates. This problem can be solved, as has been shown in Chapter 5, by considering the matrix element as one that of an operator that depends upon both electronic and nuclear displacements and by introducing a q-centroid approximation for the electronic factor. The latter is obtained as an average with DSWVO factor. The familiar Condon approximation can be so improved as to write the whole matrix element as a product of a vibrational overlap integral and an electronic factor, the latter being evaluated at some (j-centroid for the nuclear positions. 

It is not clear what the most useful applications of such procedure will turn out to be. We have already mentioned the determination of the q-centroid for evaluating... 

---